Liverpool wasted no time getting stuck into the game, taking the lead after just six minutes through Roberto Firmino.
The Brazilian forward flicked on a header from point blank range, following a lovely delivery into the box by Andrew Robertson.
Southampton issued a response almost immediately - and in similar fashion - as Che Adams found himself on the end of a James Ward-Prowse free-kick in the ninth minute to make it 1-1.
Both sides committed to attacking each other as Liverpool pressed their guests, who were keen to go forward on the counter, high up the field.
But it was the Reds who broke through, via Darwin Nunez, who finished off a superb dink over the top from England youngster Harvey Elliott in the 21st minute.
The Uruguayan would bag a brace just before half-time, despite Gavin Bazunu's best efforts, sending a superb Robertson cross goalward.
Liverpool started the second period like a team with a point to prove, creating a flurry of chances in front of Southampton's goal.
Approaching the hour mark, the Reds had slowed their pace down a notch but still dominated proceedings as the hosts attempted to claw back possession of the ball.
A few changes from the bench in the 60th minute galvanised the Saints - in particular, the introduction of Samuel Edozie - forcing several solid saves from Alisson between Liverpool's sticks.
The hosts continued to dominate proceedings going into the final 20 minutes, but Southampton firmed up at the back and were able to defend well.
Adams came close to bagging a brace of his own a couple of times in the closing stages, against the run of play, but was denied by Alisson each time as Liverpool ushered out a 3-1 win at home.
